Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.2

Time: 21.3s

Precision: 64

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

↓

\[\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2} \cdot \pi}}{t}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

C

double f(double v, double t) {
        double r188540 = 1.0;
        double r188541 = 5.0;
        double r188542 = v;
        double r188543 = r188542 * r188542;
        double r188544 = r188541 * r188543;
        double r188545 = r188540 - r188544;
        double r188546 = atan2(1.0, 0.0);
        double r188547 = t;
        double r188548 = r188546 * r188547;
        double r188549 = 2.0;
        double r188550 = 3.0;
        double r188551 = r188550 * r188543;
        double r188552 = r188540 - r188551;
        double r188553 = r188549 * r188552;
        double r188554 = sqrt(r188553);
        double r188555 = r188548 * r188554;
        double r188556 = r188540 - r188543;
        double r188557 = r188555 * r188556;
        double r188558 = r188545 / r188557;
        return r188558;
}

↓

double f(double v, double t) {
        double r188559 = 1.0;
        double r188560 = 5.0;
        double r188561 = v;
        double r188562 = r188561 * r188561;
        double r188563 = r188560 * r188562;
        double r188564 = r188559 - r188563;
        double r188565 = sqrt(r188564);
        double r188566 = r188559 * r188559;
        double r188567 = 4.0;
        double r188568 = pow(r188561, r188567);
        double r188569 = 3.0;
        double r188570 = r188569 * r188569;
        double r188571 = r188568 * r188570;
        double r188572 = r188566 - r188571;
        double r188573 = 2.0;
        double r188574 = r188572 * r188573;
        double r188575 = sqrt(r188574);
        double r188576 = atan2(1.0, 0.0);
        double r188577 = r188575 * r188576;
        double r188578 = r188565 / r188577;
        double r188579 = t;
        double r188580 = r188578 / r188579;
        double r188581 = r188565 * r188580;
        double r188582 = 3.0;
        double r188583 = pow(r188559, r188582);
        double r188584 = pow(r188562, r188582);
        double r188585 = r188583 - r188584;
        double r188586 = r188581 / r188585;
        double r188587 = r188569 * r188562;
        double r188588 = r188559 + r188587;
        double r188589 = sqrt(r188588);
        double r188590 = r188562 * r188562;
        double r188591 = r188559 * r188562;
        double r188592 = r188590 + r188591;
        double r188593 = r188566 + r188592;
        double r188594 = r188589 * r188593;
        double r188595 = r188586 * r188594;
        return r188595;
}

Error

Bits error versus v

Bits error versus t

Derivation

1. Initial program 0.4

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
2. Using strategy rm

3. Applied flip3-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}\]

4. Applied flip-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \color{blue}{\frac{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

5. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

6. Applied sqrt-div 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

7. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}} \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

8. Applied frac-times 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}}\]

9. Applied associate-/r/ 0.4

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)}\]

10. Simplified 0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{t \cdot \pi}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{\left(2 \cdot 2\right)}\right)}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]
11. Using strategy rm

12. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{t \cdot \pi}}{\color{blue}{1 \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{\left(2 \cdot 2\right)}\right)}}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

13. Applied add-sqr-sqrt 0.4

    \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}{t \cdot \pi}}{1 \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{\left(2 \cdot 2\right)}\right)}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

14. Applied times-frac 0.4

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}}{1 \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{\left(2 \cdot 2\right)}\right)}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

15. Applied times-frac 0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}{1} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{\left(2 \cdot 2\right)}\right)}}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

16. Simplified 0.3

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{\left(2 \cdot 2\right)}\right)}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

17. Simplified 0.3

    \[\leadsto \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t} \cdot \color{blue}{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]
18. Using strategy rm

19. Applied div-inv 0.3

    \[\leadsto \frac{\color{blue}{\left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{1}{t}\right)} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

20. Applied associate-*l* 0.3

    \[\leadsto \frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \left(\frac{1}{t} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}\right)}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

21. Simplified 0.2

    \[\leadsto \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \color{blue}{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2} \cdot \pi}}{t}}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

22. Final simplification 0.2

    \[\leadsto \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2} \cdot \pi}}{t}}{{1}^{3} - {\left(v \cdot v\right)}^{3}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019305 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))